We give some sufficient conditions for mappings defined on vector ultrametric spaces to be modular locally constant.

1. Introduction and Preliminaries

A metric space (X,d) in which the triangle inequality is replaced by (1.1)d(x,y)≤max⁡{d(x,z),d(z,y)},(x,y,z∈X),
is called an ultrametric space. Generalized ultrametric spaces were given in [1, 2] via partially ordered sets and some applications of them appeared in logic programming [3], computational logic [4], and quantitative domain theory [5].

In [6], the notion of a metric locally constant function on an ultrametric space was given in order to investigate certain groups of isometries and describe various Galois groups over local fields. Locally constant functions also appear in contexts such as higher ramification groups of finite extensions of Qp, and the Fontaine ring BdR+. Also, metric locally constant functions were studied in [7, 8]. On the other hand, vector ultrametric spaces are given in [9] as vectorial generalizations of ultrametrics. Hence, locally constant functions, in modular sense, can play the same role in vector ultrametric spaces as they do in usual ultrametric spaces.

In this paper, we introduce modular locally constant mappings in vector ultrametric spaces. Some sufficient conditions are given for mappings defined on vector ultrametric spaces to be modular locally constant.

We first present some basic notions.

Recall that a modular on a real linear space 𝒜 is a real valued functional ρ on 𝒜 satisfying the conditions: (1)

ρ(x)=0 if and only if x=0,

(2)

ρ(x)=ρ(-x),

(3)

ρ(αx+βy)≤ρ(x)+ρ(y), for all x,y∈𝒜 and α,β≥0, α+β=1.

Then, the linear subspace (1.2)Aρ={x∈A:ρ(αx)⟶0asα⟶0}
of 𝒜 is called a modular space.

A sequence (xn)n=1∞ in 𝒜ρ is called ρ-convergent (briefly, convergent) to x∈𝒜ρ if ρ(xn-x)→0 as n→∞, and is called Cauchy sequence if ρ(xm-xn)→0 as m,n→∞. By a ρ-closed (briefly, closed) set in 𝒜ρ we mean a set which contains the limit of each of its convergent sequences. Then, 𝒜ρ is a complete modular space if every Cauchy sequence in 𝒜ρ is convergent to a point of 𝒜ρ. We refer to [10, 11] for more details.

A cone 𝒫 in a complete modular space 𝒜ρ is a nonempty set such that

(i) 𝒫 is ρ-closed, and 𝒫≠{0};

(ii) a,b∈ℝ, a,b≥0, x,y∈𝒫⇒ax+by∈𝒫;

(iii) 𝒫∩(-𝒫)={0}, where -𝒫={-x:x∈𝒫}.

Let ⪯ be the partial order on 𝒜ρ induced by the cone 𝒫, that is, x⪯y whenever y-x∈𝒫. The cone 𝒫 is called normal if (1.3)0⪯x⪯y⟹ρ(x)≤ρ(y),(x,y∈Aρ).
The cone 𝒫 is said to be unital if there exists a vector e∈𝒫 with modular 1 such that (1.4)x⪯ρ(x)e,(x∈P).

Example 1.1.

Consider the real vector space C[0,1] consisting of all real-valued continuous functions on [0,1] equipped with the modular ρ defined by
(1.5)ρ(x)=max⁡t∈[0,1]|x(t)|2,(x∈C[0,1]).
It is not difficult to see that C[0,1] is a complete modular space and
(1.6)P={x∈C[0,1]:x(t)≥0,∀t∈[0,1]}
is a normal cone in C[0,1].

Example 1.2.

The vector space C1[0,1] consisting of all continuously differentiable real-valued functions on [0,1] equipped with the modular ρ defined by
(1.7)ρ(x)=max⁡t∈[0,1]|x(t)|+max⁡t∈[0,1]|x′(t)|,(x∈C1[0,1])
constitutes a complete modular space. The subset
(1.8)P={x∈C1[0,1]:x(t)≥0,∀t∈[0,1]}
is a unital cone in C1[0,1] with unit 1. The cone 𝒫 is not normal since, for example, x(t)=tn⪯1, for n≥1 does not imply that ρ(x)≤ρ(1).

Throughout this note, we suppose that 𝒫 is a cone in complete modular space 𝒜ρ, and ⪯ is the partial order induced by 𝒫.

Definition 1.3.

A vector ultrametric on a nonempty set 𝒳 is a mapping d:𝒳×𝒳→𝒜ρ satisfying the conditions:(CUM1)

d(x,y)≽0 for all x,y∈𝒳 and d(x,y)=0 if and only if x=y;

(CUM2)

d(x,y)=d(y,x) for all x,y∈𝒳;

(CUM3)

If d(x,z)⪯p and d(y,z)⪯p, then d(x,y)⪯p, for any x,y,z∈𝒳, and p∈𝒫.

Then the triple (𝒳,d,𝒫) is called a vector ultrametric space. If 𝒫 is unital and normal, then (𝒳,d,𝒫) is called a unital-normal vector ultrametric space.

For unital-normal vector ultrametric space (𝒳,d,𝒫), since (1.9)d(x,y)⪯ρ(d(x,y))e,d(y,z)⪯ρ(d(y,z))e,
from (CUM3) we get (1.10)d(x,z)⪯max⁡{ρ(d(x,y)),ρ(d(y,z))}e,
and therefore (1.11)ρ(d(x,z))≤max⁡{ρ(d(x,y)),ρ(d(y,z))}.
Let (𝒳,d,𝒫) be a unital-normal vector ultrametric space. If x∈𝒳 and p∈𝒫∖{0}, the ball B(x,p) centered at x with radius p is defined as (1.12)B(x,p):={y∈X:ρ(d(x,y))≤ρ(p)}.
The unital-normal vector ultrametric space (𝒳,d,𝒫) is called spherically complete if every chain of balls (with respect to inclusion) has a nonempty intersection.

The following lemma may be easily obtained.

Lemma 1.4.

Let (𝒳,d,𝒫) be a unital-normal vector ultrametric space. (1)

If a,b∈𝒳, 0⪯p and b∈B(a,p), then B(a,p)=B(b,p).

(2)

If a,b∈𝒳, 0≺p⪯q, then either B(a,p)⋂B(b,q)=∅ or B(a,p)⊆B(b,q).

Definition 1.5.

Let (𝒳,d,𝒫) be a unital-normal vector ultrametric space. A mapping f:𝒳→𝒫∖{0} is said to be modular locally constant provided that for any x∈𝒳 and any y∈B(x,f(x)) one has ρ(f(x))=ρ(f(y)).

2. Main TheoremTheorem 2.1.

Let (𝒳,d,𝒫) be a spherically complete unital-normal vector ultrametric space and T:𝒳→𝒳 be a mapping such that for every x,y∈𝒳,x≠y, either
(2.1)ρ(d(Tx,Ty))<max⁡{ρ(d(x,Tx)),ρ(d(y,Ty))}
or
(2.2)ρ(d(Tx,Ty))≤ρ(d(x,y)).
Then there exists a subset B of 𝒳 such that T:B→B and the mapping
(2.3)f(x)=d(x,Tx),(x∈B)
is modular locally constant.

Proof.

Let ℰ={Ba}a∈𝒳 where Ba=B(a,d(a,Ta)). Consider the partial order ⊑ on ℰ defined by
(2.4)Ba⊑BbiffBb⊆Ba,
where a,b∈𝒳. If ℰ1 is any chain in ℰ, then the spherical completeness of (𝒳,d,𝒫) implies that the intersection Ω of elements of ℰ1 is nonempty.

Suppose that (2.1) holds. Let b∈Ω and Ba∈ℰ1. Obviously b∈Ba, so ρ(d(a,b))≤ρ(d(a,Ta)). For any x∈Bb, we have(2.5)ρ(d(x,b))≤ρ(d(b,Tb))≤max⁡{ρ(d(b,a)),ρ(d(a,Ta)),ρ(d(Ta,Tb))}<max⁡{ρ(d(b,a)),ρ(d(a,Ta)),max⁡{ρ(d(a,Ta)),ρ(d(b,Tb))}}(by(2.1))≤max⁡{ρ(d(b,a)),ρ(d(a,Ta)),ρ(d(b,Tb))}≤max⁡{ρ(d(a,Ta)),ρ(d(b,Tb))}=ρ(d(a,Ta)),ρ(d(x,a))≤max⁡{ρ(d(x,b)),ρ(d(b,a))}≤ρ(d(a,Ta)).
So for every Ba∈ℰ1, Bb⊆Ba; that is, Bb is an upper bound in ℰ for the family ℰ1. By Zorn's lemma, there exists a maximal element in ℰ1, say Bz. If b∈Bz, ρ(d(b,z))≤ρ(d(z,Tz)), and we get
(2.6)ρ(d(b,Tb))≤max⁡{ρ(d(b,z)),ρ(d(z,Tz)),ρ(d(Tz,Tb))}<max⁡{ρ(d(b,z)),ρ(d(z,Tz)),max⁡{ρ(d(z,Tz)),ρ(d(b,Tb))}}(by(2.1))≤max⁡{ρ(d(b,z)),ρ(d(z,Tz)),ρ(d(b,Tb))}≤max⁡{ρ(d(z,Tz)),ρ(d(b,Tb))}=ρ(d(z,Tz)).
Then
(2.7)ρ(d(b,Tb))≤ρ(d(z,Tz)).
Since b∈Bb∩Bz, we have Bb⊆Bz by Lemma 1.4. But Tb∈Bb, so T:Bz→Bz. Now we show that ρ(f(b))=ρ(f(z)) for every b∈Bz. It is clear that ρ(d(b,Tb))≤ρ(d(z,Tz)), for all b∈Bz. Suppose ρ(d(b,Tb))<ρ(d(z,Tz)) for some b∈Bz. We have ρ(d(b,z))≤ρ(d(z,Tz)), and
(2.8)ρ(d(z,Tz))≤max⁡{ρ(d(z,b)),ρ(d(b,Tb)),ρ(d(Tb,Tz))}<max⁡{ρ(d(b,z)),ρ(d(b,Tb)),max⁡{ρ(d(b,Tb)),ρ(d(z,Tz))}}(by(2.1))≤max⁡{ρ(d(b,z)),ρ(d(b,Tb)),ρ(d(z,Tz))}≤max⁡{ρ(d(b,Tb)),ρ(d(z,Tz))}=ρ(d(z,Tz)).
which is a contradiction. Thus f is modular locally constant on Bz.

Suppose that (2.2) holds. As above, let b∈Ω and Ba∈ℰ1. Obviously b∈Ba, so ρ(d(a,b))≤ρ(d(a,Ta)). For any x∈Bb, we have(2.9)ρ(d(x,b))≤ρ(d(b,Tb))≤max⁡{ρ(d(b,a)),ρ(d(a,Ta)),ρ(d(Ta,Tb))}≤max⁡{ρ(d(b,a)),ρ(d(a,Ta))}(by(2.2))=ρ(d(a,Ta)).
Thus
(2.10)ρ(d(x,a))≤max⁡{ρ(d(x,b)),ρ(d(b,a))}≤ρ(d(a,Ta)),ρ(d(x,a))≤max⁡{ρ(d(x,b)),ρ(d(b,a))}≤ρ(d(a,Ta)).
So, for every Ba∈ℰ1, Bb⊆Ba; that is, Bb is an upper bound for the family ℰ1. Again, by Zorn's lemma there exists a maximal element in ℰ1, say Bz. For any b∈Bz, we have
(2.11)ρ(d(b,Tb))≤max⁡{ρ(d(b,z)),ρ(d(z,Tz)),ρ(d(Tz,Tb))}≤max⁡{ρ(d(b,z)),ρ(d(z,Tz)),ρ(d(z,b))}(by(2.2))=ρ(d(z,Tz)).
This implies that b∈Bb∩Bz, and Lemma 1.4 gives Bb⊆Bz. Since Tb∈Bb, so T:Bz→Bz.

If z=Tz, then f(x)=0 on Bz and this yields the result. If z≠Tz, we show that ρ(f(b))=ρ(f(z)) for every b∈Bz. Since ρ(d(b,Tb))≤ρ(d(z,Tz)) for any b∈Bz, let us suppose that for some b∈Bz, ρ(d(b,Tb))<ρ(d(z,Tz)). So ρ(d(b,z))≤ρ(d(z,Tz)) and(2.12)ρ(d(z,Tz))≤max⁡{ρ(d(z,b)),ρ(d(b,Tb)),ρ(d(Tb,Tz))}≤max⁡{ρ(d(b,z)),ρ(d(b,Tb)),ρ(d(z,b))}(by(2.2))=ρ(d(b,z)),
thus ρ(d(b,z))=ρ(d(z,Tz)). But ρ(d(b,z))=ρ(d(z,Tz))>ρ(d(b,Tb)) implies that z∈Bz, but z∉Bb and hence Bb⊊Bzwhich contradicts the maximality of Bz. This completes the proof.

In the following, we assume that (𝒳,d,𝒫) is a spherically complete unital-normal vector ultrametric space.

Corollary 2.2.

Let T:𝒳→𝒳 be a mapping such that for all x,y∈𝒳,x≠y,
(2.13)ρ(d(Tx,Ty))<max⁡{ρ(d(y,Tx)),ρ(d(x,Ty))}.
Then there exists a subset B of 𝒳 such that T:B→B and the mapping f defined in (2.3) is modular locally constant.

Proof.

Since
(2.14)ρ(d(y,Tx))≤max⁡{ρ(d(y,x)),ρ(d(x,Tx))},ρ(d(x,Ty))≤max⁡{ρ(d(x,y)),ρ(d(y,Ty))},
for all x,y∈𝒳,x≠y, we get
(2.15)ρ(d(x,y))≤max⁡{ρ(d(x,Tx)),ρ(d(Tx,Ty)),ρ(d(Ty,y))}
for all x,y∈𝒳,x≠y. Now, if
(2.16)max⁡{ρ(d(x,Tx)),ρ(d(y,Ty))}<ρ(d(Tx,Ty)),
then
(2.17)ρ(d(Tx,Ty))<max⁡{ρ(d(y,Tx)),ρ(d(x,Ty))}(by(2.13))≤max⁡{ρ(d(x,y)),ρ(d(x,Tx)),ρ(d(y,Ty))}(by(2.14))≤max⁡{ρ(d(x,Tx)),ρ(d(Tx,Ty)),ρ(d(Ty,y))}=ρ(d(Tx,Ty)),(by(2.16))
which is a contradiction. Thus ρ(d(Tx,Ty))≤max{ρ(d(x,Tx)),ρ(d(y,Ty))}, and so
(2.18)ρ(d(x,y))≤max⁡{ρ(d(x,Tx)),ρ(d(y,Ty))}.
Therefore
(2.19)ρ(d(Tx,Ty))<max⁡{ρ(d(y,Tx)),ρ(d(x,Ty))},(by(2.13))≤max⁡{ρ(d(x,y)),ρ(d(x,Tx)),ρ(d(y,Ty))}(by(2.14))≤max⁡{ρ(d(x,Tx)),ρ(d(y,Ty))}.(by(2.18))
Now, Theorem 2.1 completes the proof.

Corollary 2.3.

Let T:𝒳→𝒳 be a mapping such that for all x,y∈𝒳,x≠y,
(2.20)ρ(d(Tx,Ty))<ρ(d(x,y)).
Then there exists a subset B of 𝒳 such that T:B→B and the mapping f defined in (2.3) is modular locally constant.

Proof.

We have
(2.21)ρ(d(x,y))≤max⁡{ρ(d(x,Tx)),ρ(d(Tx,Ty)),ρ(d(Ty,y))}<max⁡{ρ(d(x,y)),ρ(d(x,Tx)),ρ(d(Ty,y))}(by(2.20))≤max⁡{ρ(d(x,Tx)),ρ(d(y,Ty))},
for all x,y∈𝒳,x≠y. Again, Theorem 2.1, completes the proof.

Acknowledgments

The authors would like to thank the referee for his/her valuable comments on this paper. The second author’s research was in part supported by a grant from IPM (No. 89470128).